First example: the integers
One of the more familiar groups is the set of integers
• For all integers $${\displaystyle a}$$, $${\displaystyle b}$$ and $${\displaystyle c}$$, … See more

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group … See more

Examples and applications of groups abound. A starting point is the group $${\displaystyle \mathbb {Z} }$$ of integers with addition as … See more

An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set See more

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations See more

When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses … See more

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups See more